Maxima and Minima

01 Rectangle of maximum perimeter inscribed in a circle

Problem 01
Find the shape of the rectangle of maximum perimeter inscribed in a circle.

Maxima and Minima Using Trigonometric Functions

Many problems in application of maxima and minima may be solved easily by making use of trigonometric functions. The basic idea is the same; identify the constant terms and identify the variable to be maximized or minimized, differentiate that variable then equate to zero.

28-29 Time Rates: Two cars driving on roads that intersects at 60 degree

28-figure-48.jpgProblem 28
At noon a car drives from A (Fig. 48) toward C at 60 miles per hour. Another car starting from B at the same time drives toward A at 30 miles per hour. If AB = 42 miles, find when the cars will be nearest each other.

22-24 One car from a city starts north, another car from nearby city starts east

Problem 22
One city C, is 30 miles north and 35 miles east from another city, D. At noon, a car starts north from C at 40 miles per hour, at 12:10 PM, another car starts east from D at 60 miles per hour. Find when the cars will be nearest together.

72 - 74 Light intensity of illumination and theory of attraction

Problem 72
A light is to be placed above the center of a circular area of radius a. What height gives the best illumination on a circular walk surrounding the area? (When light from a point source strikes a surface obliquely, the intensity of illumination is

$I = \dfrac{k \sin \theta}{d^2}$

where θ is the angle of incidence and d the distance from the source.)

69 - 71 Shortest and most economical path of motorboat

Problem 69
A man on an island 12 miles south of a straight beach wishes to reach a point on shore 20 miles east. If a motorboat, making 20 miles per hour, can be hired at the rate of \$2.00 per hour for the time it is actually used, and the cost of land transportation is \$0.06 per mile, how much must he pay for the trip?

66 - 68 Maxima and minima: Pyramid inscribed in a sphere and Indian tepee

Problem 66
Find the largest right pyramid with a square base that can be inscribed in a sphere of radius a.

64 - 65 Maxima and minima: cone inscribed in a sphere and cone circumscribed about a sphere

Problem 64
A sphere is cut to the shape of a circular cone. How much of the material can be saved? (See Problem 63).

62 - 63 Maxima and minima: cylinder inscribed in a cone and cone inscribed in a sphere

Problem 62
Inscribe a circular cylinder of maximum convex surface area in a given circular cone.

60 - 61 Maxima and minima problems of a folded page

Figure 41Problem 60
One corner of a leaf of width a is folded over so as just to reach the opposite side of the page. Find the width of the part folded over when the length of the crease is a minimum. See Figure 41.


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